Bradford's season-ending knee injury last weekend prompted speculation the Rams would try to sign Tim Tebow. Instead, they reached out to Favre's agent, Bus Cook, ESPN said.
Favre, 44, who last played in 2010, said he was not interested in coming out of retirement.
The Rams signed veteran quarterback Brady Quinn, who was released by the New York Jets earlier this week, and brought back Austin Davis, who spent the 2012 and 2013 preseason with the Rams. Quinn and Davis will be backups to Kellen Clemens, whose first start of the season will be Monday against the Seattle Seahawks.